how to move the camera without making the reference background image disappear?
You can use an empty as image



Select the image in the properties->Empty->Display tab



You'll get your image (or video) in an empty. The advantage of this method is that your image will be visible as you rotate, and is visible even in when the 3D viewport shading is set to wireframe or bounding box.

Note that empties will not be rendered on the final render...